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On geodesic exponential maps of the Virasoro group. (English) Zbl 1121.35111
Summary: We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics ${\mu }^{\left(k\right)}$ $\left(k\ge$ 0) on the Virasoro group $Vir$ and show that for $k\ge 2$, but not for $k=0,1$, each of them defines a smooth Fréchet chart of the unital element $e\in$ Vir. In particular, the geodesic exponential map corresponding to the Korteweg-de Vries (KdV) equation $\left(k=0\right)$ is not a local diffeomorphism near the origin.
MSC:
 35Q35 PDEs in connection with fluid mechanics 37K30 Relations of infinite-dimensional systems with algebraic structures 37K25 Relations of infinite-dimensional systems with differential geometry 58B25 Group structures and generalizations on infinite-dimensional manifolds
References:
 [1] Arnold V. (1966). Sur la geometrié differentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluids parfaits. Ann. Inst. Fourier 16(1):319–361 [2] Arnold V., Khesin B. (1998). Topological Methods in Hydrodynamics. Springer–Verlag, New York [3] Bona, J.-L., Smith, R.: The initial-value problem for Korteweg–de Vries equation. Phil. Trans. R. Soc. Lon. Ser. A, Math. Phys. Sci. 278, 555–601 (1975) [4] Burgers, J.: A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1, 171–199 (1948) [5] Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993) [6] Constantin, A.: On the Cauchy problem for the periodic Camassa–Holm equation. J. Differ. Eq. 141, 218–235 (1997) [7] Constantin, A., Escher, J.: Well-posedness, global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 51, 475–504 (1998) [8] Constantin, A., Escher, J.: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math. Z. 233, 75–91 (2000) [9] Constantin, A., McKean, H.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52, 949–982 (1999) [10] Constantin, A., Kolev, B.: On the geometric approach to the motion of inertial mechanical systems. J. Phys. A 35, R51–R79 (2002) [11] Constantin, A., Kolev, B.: Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78, 787–804 (2003) [12] Constantin, A., Kolev, B., Lenells, J.: Integrability of invariant metrics on the Virasoro group. Phys. Lett. A 350, 75–80 (2006) [13] De Lellis, C., Kappeler, T., Topalov, P.: Low regularity solutions of the Camassa–Holm equation (to appear in Comm. in PDE) [14] Ebin, D., Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92, 102–163 (1970) [15] Fokas, A., Fuchssteiner, B.: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Physica D 4, 47–66 (1981) [16] Hamilton, R.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. 7, 66–222 (1982) [17] Holm, D., Marsden, J., Ratiu, T.: The Euler–Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137, 1–81 (1998) [18] Kappeler, T., Pöschel, J.: KdV&KAM. Springer-Verlag, Berlin (2003) [19] Kappeler, T., Topalov, P.: Well-posedness of KdV on H 1 ( $𝕋$ ) (to appear in Duke Math. J.) [20] Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58, 181–205 (1975) [21] Khesin, B., Misiolek, G.: Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176, 116–144 (2003) [22] Kopell, N.: Commuting diffeomorphisms. (Proc. Symp. Pure Math). Am. Math. Soc. 14, 165–184 (1970) [23] Kouranbaeva, S.: The Camassa–Holm equation as a geodesic flow on the diffeomorphism group. J. Math. Phys. 40, 857–868 (1999) [24] Lang, S.: Differential manifolds. Addison-Wesley Series in Mathematics, Mass (1972) [25] Lenells, J.: The correspondence between KdV and Camassa–Holm. Int. Math. Res. Not. 71, 3797–3811 (2004) [26] McKean, H.P.: The Liouville correspondence between the Korteweg–de Vries and the Camassa–Holm hierarchies. Comm. Pure Appl. Math. 56, 998–1015 (2003) [27] Michor, P., Ratiu, T.: On the geometry of the Virasoro–Bott group. J. Lie Theory 8, 293–309 (1998) [28] Milnor, J.: Remarks on infinite-dimensional Lie groups. Les Houches, Session XL, 1983, Elsevier Science Publishers B.V. (1984) [29] Misiolek, G.: A shallow water equation as a geodesic flow on the Bott–Virasoro group. J. Geom. Phys. 24, 203–208 (1998) [30] Misiolek G. (2002). Classical solutions of the periodic Camassa–Holm equation. GAFA 12, 1080–1104 · Zbl 1158.37311 · doi:10.1007/PL00012648 [31] Ovsienko V., Khesin B. (1987). Korteweg–de Vries superequations as an Euler equation. Funct. Anal. Appl. 21, 81–82 [32] Quantum fields and strings: a course for mathematicians. vols. 1 and 2. Deligne, P., Etingof, P., etc., American Mathematical Society, Providence, RI; Institute for Advanced Study, Princeton, NJ (1999)